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Stable rank for crossed products by finite group actions with the weak tracial Rokhlin property

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arxiv 2407.09867 v1 pith:SMN2ZANV submitted 2024-07-13 math.OA

classification math.OA
keywords propertyalphafinitegrouprankrokhlinstabletracial
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abstract

Let $A$ be an infinite-dimensional stably finite simple unital C*-algebra, let $G$ be a finite group, and let $\alpha\colon G\rightarrow \mathrm{Aut}(A)$ be an action of $G$ on $A$ which has the weak tracial Rokhlin property. We prove that if $A$ has property (TM), then the crossed product $A\rtimes_\alpha G$ has property (TM). As a corollary, if $A$ is an infinite-dimensional separable simple unital C*-algebra which has stable rank one and strict comparison, $\alpha\colon G\rightarrow \mathrm{Aut}(A)$ is an action of a finite group $G$ on $A$ with the weak tracial Rokhlin property, then $A\rtimes_\alpha G$ has stable rank one.

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  1. Uniform property $\Gamma$ for Crossed products by group actions with the Rokhlin-type properties

    math.OA 2024-12 conditional novelty 5.0 of 10

    Finite group actions with the weak tracial Rokhlin property and compact group actions with the tracial Rokhlin property with comparison preserve uniform property Gamma in crossed products and fixed-point algebras.

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